Mac laptops don't have dial-up modems, so, when I tried to buy external one (dongle) in a Brisbane City Next Byte Mac store found out they don't have such dongle...

Why not try a 3G modem instead? Telstra, Optus and Vodafone all sell prepaid modems. The advantages are that they just plug into the USB port – I'm using one now to post this from my MacBook Pro – and they're broadband, not slow, old dial-up!